A Lower Bound on the Sinc Function and Its Application

A lower bound on the sinc function is given. Application for the sequence {b n}n=1 ∞ which related to Carleman inequality is given as well.


Introduction
The sinc function is defined to be This function plays a key role in many areas of mathematics and its applications [1][2][3][4][5][6].
The following result that provides a lower bound for the sinc is well known as Jordan inequality [7]: where equality holds if and only if = /2. This inequality has been further refined by many authors in the past few years .
In [36], it was proposed that We noticed that the lower bound in (3) is the fractional function. Similar result has been reported as follows [1]: To the best of the authors' knowledge, few results have been obtained on fractional lower bound for the sinc function. It is the first aim of the present paper to establish the following fractional lower bound for the sinc function.
In [37], Yang proved that for any positive integer , the following Carleman type inequality holds: From a mathematical point of view, the sequence { } ∞ =1 has very interesting properties. Yang [38] and Gyllenberg and Ping [39] have proved that, for any positive integer , > 0, . 2 The Scientific World Journal In [40], the authors proved that where As an application of Theorem 1, it is the second aim of the present paper to give a better upper bound on the sequence Theorem 2. For any positive integer ⩾ 2, one has

The Proof of Theorem 1
The proof is not based on (3). We first prove the following result.

The Proof of Theorem 2
First, we need an auxiliary result.
Taking the natural log gives Taking the second derivative of both sides of (38), we have By Lemma 4, it follows that Thus, and therefore for 0 ⩽ ⩽ 1/2, we have For the case 1/2 < ⩽ 1, since (1/2) = 2/ , (1) = 1, and is concave up, it follows that This proves Theorem 2.

Conflict of Interests
There is no conflict of interests regarding the publication of this paper.